Applications of Integration

The function has positive values for some $x$ in the interval.

The function has negative values for some $x$ in the interval.

The function is always negative over the interval.

The function is always positive over the interval.

The function is discontinuous.

The function is constant over the interval.

The function is decreasing over the interval.

The function is increasing over the interval.

The function has no x-intercepts.

The function is symmetric about the x-axis.

The function is increasing over the interval.

The function is continuous over the interval.

There exists a number $c$ in $(-4,6)$ such that $h(c)=h(-4)+h(6)$.

$h(p)$'s maximum and minimum must occur at $p = -4$ and $p =6$ respectively.

$h(p)$ may have symmetry around $p = 1$ or be constant on $[-4,6]$.

The derivative $h'(p)$ has to be zero at least once in every subinterval within $[-4,6]$.

The average value becomes zero.

The average value is also multiplied by the same constant.

The average value remains the same.

The average value is divided by the constant.

The function is always decreasing over the interval.

The function has negative values for some x-value in the interval.

The function is always increasing over the interval.

The function has no x-intercepts.

$\int_{a}^{b} f(x) dx$

$\frac{1}{2}(f(a)+f(b))$

$\frac{f(b)-f(a)}{b-a}$

$\frac{1}{b-a} \int_{a}^{b} f(x) dx$

$\int_c^d g'(u) du$

$(d - c)\cdot g'\left(\frac{c+d}{2}\right)$

$\frac{g(d)-g(c)}{d-c}$

$\frac{\int_c^d g(u)du}{d-c}$

$\frac{1}{d-c} \int_c^d h(x)\ dx$

$( h(d)+h(c) ) \cdot \frac{(d-c)}{2}$

$( h(d)-h(c) ) \cdot \frac{(d+c)}{2}$

$\int_c^d h'(x)\ dx + h(c)$

The average value of $f(x)$ is not necessarily equal to any specific function value.

The average value of $f(x)$ is always equal to $f(a)$.

The average value of $f(x)$ is always equal to the sum of $f(a)$ and $f(b)$ divided by 2.

The average value of $f(x)$ is always equal to $f(b)$.